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When catalysis is useful for probabilistic entanglement transformation
Yuan Feng,1, ∗ Runyao Duan,1, † and Mingsheng Ying1, ‡
1
State Key Laboratory of Intelligent Technology and Systems,
Department of Computer Science and Technology Tsinghua University, Beijing, China, 100084
We determine all 2 × 2 quantum states that can serve as useful catalysts for a given probabilistic entanglement transformation, in the sense that they can increase the maximal transformation
probability. When higher-dimensional catalysts are considered, a sufficient and necessary condition
is derived under which a certain probabilistic transformation has useful catalysts.
PACS numbers: 03.67.Mn,03.65.Ud
In the field of quantum information theory, entanglement plays an essential role in quantum information processing such as quantum cryptography [1], quantum superdense coding [2] and quantum teleportation [3]. When
entanglement is treated as a type of resource, the study of
transformations between different forms of entanglement
becomes very crucial. It is well known that the entanglement quantity shared among separate parties cannot
be increased only using local operations on the separate
subsystems and classical communication between them
(or LOCC for short). The restriction on the possibility of entanglement transformations that can be realized
by LOCC is, however, beyond this. Nielsen proved in his
brilliant work [4] that a pure bipartite entangled quantum
state |ψ1 i can be transformed into another pure bipartite
entangled state |ψ2 i by LOCC if and only if λψ1 ≺ λψ2 ,
where the probability vectors λψ1 and λψ2 denote the
Schmidt coefficient vectors of |ψ1 i and |ψ2 i, respectively.
Here the symbol ≺ stands for the “majorization relation”. An n-dimensional probability vector x is said to
be majorized by another n-dimensional probability vector y, denoted by x ≺ y, if the following relations hold
l
X
i=1
x↓i ≤
l
X
yi↓
for any
1 ≤ l < n,
i=1
where x↓ denotes the vector obtained by rearranging the
components of x in nonincreasing order.
What Nielsen has done indeed gives a connection between the theory of majorization in linear algebra [5] and
the entanglement transformation. Furthermore, since the
sufficient and necessary condition is very easy to check,
it is extremely useful to decide whether one pure bipartite entangled state can be transformed into another pure
bipartite state by LOCC. There exist, however, incomparable states in the sense that any one cannot be transformed into another only using LOCC. To cope with the
transformation between incomparable states, Vidal [6]
generalized Nielsen’s work with a probabilistic manner.
He found that although a deterministic transformation
∗ Electronic
address: [email protected]
address: [email protected]
‡ Electronic address: [email protected]
† Electronic
cannot be realized between incomparable states, a probabilistic one is always possible. Furthermore, he gave an
explicit expression of the maximal probability of transforming one state to another. To be more specific, let
P (|ψi → |φi) denote the maximal transformation probability of transforming |ψi into |φi by LOCC; then
P (|ψi → |φi) = min
1≤l≤n
El (λψ )
,
El (λφ )
where n is the maximum of the Schmidt numbers of |ψi
Pn
and |φi, and El (x) denotes the abbreviation of i=l x↓i
for probability vector x.
Another interesting phenomenon was discovered by
Jonathan and Plenio [7] that sometimes an entangled
state can help in making impossible entanglement transformations into possible without being consumed at all.
That is, there exist quantum states |ψ1 i, |ψ2 i, and |φi
such that |ψ1 i 9 |ψ2 i but |ψ1 i ⊗ |φi → |ψ2 i ⊗ |φi. In
this transformation, the role of the state |φi is just like a
catalyst in a chemical process. They found by examining
an example that in some cases an appropriately chosen
catalyst can increase the maximal transformation probability of incomparable states. It was also shown that
enhancement of the maximal transformation probability
is not always possible. However, there were no further
results about such an interesting field in their paper.
In this paper, we examine the ability of catalysts in a
probabilistic entanglement transformation. We first consider the simple case of when a given probabilistic transformation has useful 2 × 2 catalysts and determine all of
them. Then a sufficient and necessary condition is derive
which can decide whether or not a certain transformation
has (not necessarily 2 × 2) useful catalysts.
For simplicity, in what follows we denote a quantum
state by the probability vector of its Schmidt coefficients. This will not cause any confusion because it is
well known that the fundamental properties of a bipartite quantum state under LOCC are completely determined by its Schmidt coefficients. Therefore, from now
on, we consider only probability vectors instead of quantum states and always identify a probability vector with
the quantum state represented by it.
Suppose x, y are two n-dimensional probability vectors and the components are arranged nonincreasingly.
It is well known that if P (x → y) = En (x)/En (y) or
P (x → y) = E1 (x)/E1 (y)(= 1), then the maximal prob-
2
ability of transforming x into y cannot be increased by
any catalyst. That is, for any probability vector c, we
have P (x ⊗ c → y ⊗ c) = P (x → y). Thus, in what
follows, we assume
P (x → y) <
E1 (x)
En (x)
, P (x → y) <
.
En (y)
E1 (y)
(1)
Without loss of generality, we concentrate on catalysts
with nonzero components, since c and c⊕0 have the same
catalysis ability for any probability vector c in the sense
that in any situation, if one serves as a partial catalyst
for some transformation, so does the other for the same
transformation. Let
L = {l : 1 < l < n and P (x → y) =
El (x)
}.
El (y)
In what follows, we derive a sufficient and necessary condition when a two-dimensional catalyst can increase the
maximal transformation probability from x to y. In order
to state the theorem compactly, we first denote
mrr21
xr −1 yr −1
= min{ 1 , 1 }
x r2
yr2
Here ri , 1 ≤ ri ≤ n + 1, denotes the smallest index of the
components of x in the summands of El (x ⊗ c) that have
the form ci xj , where 1 ≤ j ≤ n. The case ri = n + 1
denotes that any term that has the form ci xj does not
occur. In the case of repeated values of components of
x ⊗ c, we regard terms with larger i, and larger j if they
have the same i, to be included in the sum first.
From these assumptions, we can show that r1 ≥ r2 .
Otherwise, r1 ≤ r2 − 1, and from the fact c1 xr1 is in
the summands of El (x ⊗ c) while c2 xr2 −1 is not, we can
deduce that c1 xr1 ≤ c2 xr2 −1 . But on the other hand, we
have c1 ≥ c2 and xr1 ≥ xr2 −1 . So it follows that c1 = c2
and xr1 = xr2 −1 . Especially, c1 xr2 −1 = c2 xr2 −1 which
contradicts our assumption that the term with larger i is
included in El (x ⊗ c) first, since the former is while the
latter is not included in sx . Furthermore, from Eq. (6)
we have r1 + r2 = l + 1; then, r2 < n + 1 since 1 < l ≤ 2n.
Now, by the definition of El (y ⊗ c) and P (x → y), the
following inequality is easy to check:
c1
El (x ⊗ c)
≥
El (y ⊗ c)
(2)
and
c1
n
X
i=r1
n
X
x i + c2
yi + c2
i=r1
Mrr12 = max{
x r1
y r1
,
}
xr2 −1 yr2 −1
P (x → y)(c1
is not empty, where the intersection is taken over all pairs
of r1 , r2 such that r1 , r2 ∈ L ∪ {n + 1}, r1 ≥ r2 , and
r2 < n + 1. In fact, any two-dimensional probability
vector (c1 , c2 ) with c1 ≥ c2 can serve as a useful catalyst
for transforming x into y if and only if c2 /c1 ∈ S.
Proof. Suppose c = (c1 , c2 ) is a 2-dimensional probability vector and c1 ≥ c2 . We need only show that c
cannot serve as a useful catalyst for transforming x into
y, that is,
i=r2
n
X
xi
yi
i=r2
(3)
r2
for any r1 , r2 ∈ L. Furthermore, we let Mn+1
= 0.
Theorem 1. The maximal probability of transforming
x into y can be increased by a 2-dimensional catalyst if
and only if the set
\
S = {(0, Mrr12 ) ∪ (mrr21 , 1)}
(4)
n
X
n
X
y i + c2
i=r1
≥
n
X
c1
y i + c2
i=r1
c2
≤ mrr21 ,
c1
n
X
El (x ⊗ c) = c1
n
X
i=r1
x i + c2
n
X
i=r2
xi .
(6)
yi
xi + c2
n
X
xi
(8)
yi .
(9)
i=r2
and
El (y ⊗ c) = c1
n
X
i=r1
where any constraint having meaningless terms is satisfied automatically.
For an arbitrarily fixed integer l satisfying 1 < l ≤ 2n,
we can arrange the summands in El (x ⊗ c) as
i=r2
n
X
The
Pnfirst equality
Pn holds if and only if El (y ⊗ c) =
c1 i=r1 yi + c2 i=r2 yi while the second equality holds
if and only if r1 and r2 are both included in L or,
r2 ∈ L and r1 = n + 1. Notice that l and ri can
be uniquely determined by each other from Eq. (6);
it follows that the sufficient and necessary condition of
when P (x ⊗ c → y ⊗ c) = P (x → y) is there exist
r1 , r2 ∈ L ∪ {n + 1} satisfying r1 ≥ r2 and r2 < n + 1,
such that
El (x ⊗ c) = c1
(5)
(7)
i=r2
i=r1
Mrr12 ≤
yi )
= P (x → y).
P (x ⊗ c → y ⊗ c) = P (x → y)
if and only if there exist r1 , r2 ∈ L ∪ {n + 1}, r1 ≥ r2 ,
and r2 < n + 1, such that
n
X
y i + c2
n
X
i=r2
In what follows, we derive the conditions presented in
Eq. (5) from Eqs.(8) and (9). In fact, what Eq. (8) says
is simply that c1 xr1 −1 ≥ c2 xr2 and c2 xr2 −1 ≥ c1 xr1 or,
equivalently,
c2
xr −1
x r1
≤
≤ 1 .
xr2 −1
c1
x r2
(10)
3
The special case when r1 takes value n+1 can be included
in Eq. (10) simply by assuming that the constraints in
Eq. (10) containing meaningless terms are automatically
satisfied. Analogously, we can show that Eq. (9) is equivalent to
c2
yr −1
yr1
≤
≤ 1 .
(11)
yr2 −1
c1
yr2
Combining Eqs.(10) and (11) together and noticing the
denotations in Eqs.(2) and (3), we derive the sufficient
and necessary condition for two-dimensional probability
vector c such that P (x ⊗ c → y ⊗ c) = P (x → y) is just
what Eq. (5) presents. That completes our proof.
A special and perhaps more interesting case of the
above theorem is when the number of elements in L is 1,
that is L = {l} for some 1 < l < n. In this case, the possible values of the pair (r1 , r2 ) are just (l, l) and (n + 1, l).
So the set S in Eq. (4) is simply (0, Mll ) ∩ (mln+1 , 1)
and the sufficient and necessary condition of when twodimensional catalysts exist which can increase the maximal probability of transforming x into y is mln+1 < Mll ,
that is,
min{
xn yn
xl
yl
, } < max{
,
}.
xl yl
xl−1 yl−1
(12)
Furthermore, any two-dimensional probability vector
c, c1 ≥ c2 , which satisfies
min{
xn yn
c2
xl
yl
, }<
< max{
,
},
xl yl
c1
xl−1 yl−1
(13)
can be a useful catalyst for this transformation. In the
simplest case of n = 3 (notice that when n = 2, any
entanglement transformation has no catalyst), the set L
must be {2} and furthermore, from the assumption Eq.
(1) we have x3 /x2 > y3 /y2 and x2 /x1 < y2 /y1 . It follows
that when x and y are both three-dimensional, the sufficient and necessary condition for them to have a useful
two-dimensional catalyst is
y3
y2
<
y2
y1
(14)
and any c = (c1 , c2 ) with c1 ≥ c2 and
c2
y2
y3
<
<
y2
c1
y1
(15)
can increase the maximal transformation probability.
Note that these conditions are all irrelevant to x.
To illustrate the utility of the above theorem, let us
give some simple examples.
Example 1. This example is given by Jonathan and
Plenio in [7]. Let x = (0.6, 0.2, 0.2) and y = (0.5, 0.4, 0.1),
we have y3 /y2 = 0.25 and y2 /y1 = 0.8. So from Eq. (15),
any state c = (c1 , c2 ), c1 ≥ c2 , can serve as a useful
catalyst for transforming x into y, provided that 0.25 <
c2 /c1 < 0.8 or, equivalently, 5/9 < c1 < 4/5. Especially,
when choosing c1 = 0.65, we get c = (0.65, 0.35), which
is the one given in [7].
Suppose x is just as above while y = (0.5, 0.3, 0.2);
then, y3 /y2 = 2/3 and y2 /y1 = 0.6. Since 0.6 < 2/3, we
deduce that any two-dimensional state cannot serve as a
useful catalyst for the probabilistic transformation from
x to y in the sense that it cannot increase the maximal
transformation probability.
Example 2. This well-known example is exactly
the original one that Jonathan and Plenio used to
demonstrate entanglement catalysis [7].
Let x =
(0.4, 0.4, 0.1, 0.1) and y = (0.5, 0.25, 0.25, 0); then L =
{3}, and from (12) we have
min{
x4 y4
, } = min{1, 0} = 0
x3 y3
and
max{
x3 y3
, } = max{0.25, 1} = 1.
x2 y2
It follows from Eq. (13) that any state c = (c1 , c2 ),
0 < c2 /c1 < 1, can serve as a useful catalyst. That is,
any two-dimensional nonpure and nonuniform state can
increase the maximal transformation probability from x
to y.
We have examined when there exists a two-dimensional
catalyst which is useful for probabilistic transformation. In what follows, we consider the case of higherdimensional catalysts and derive a sufficient and necessary condition for a certain probabilistic transformation
to have a useful (not necessarily two-dimensional) catalyst. More important, the proof process indeed constructs an appropriate catalyst explicitly. Some techniques in the proof are from Lemma 4 in [8].
Theorem 2. Suppose x and y are two n-dimensional
probability vectors with the components ordered nonincreasingly. Then there exists a probability vector c such
that P (x ⊗ c → y ⊗ c) > P (x → y) if and only if
P (x → y) < min{xn /yn , 1}.
Proof. The “only if” part is easy and we omit the
details here. The proof of “if” is as follows.
We denote P (x → y) as P for simplicity in this proof.
Let h, 1 ≤ h < n, be the smallest index of the components such that xh /yh 6= P and α be a positive real
number such that P yn /xn < α < 1. Furthermore, if
P > xh /yh , then assume α > xh /(P yh ); otherwise, assume α > P yh /xh . Let k be a positive integer such that
xn > xh αk−1 and
c = (1, α, α2 , . . . , αk−1 ).
We omit the normalization of c here. In what follows, we
show that for any 1 < l < nk, El (x ⊗ c) > P El (y ⊗ c);
then, the catalyst we constructed above indeed increases
the maximal transformation probability.
Fix l as an arbitrary integer that satisfies 1 < l < nk
and denote sx = El (x ⊗ c). It is obvious that we can
4
arrange the summands in sx as
sx =
n k−1
X
X
xi α j .
i=1 j=ri
Here ri , 0 ≤ ri ≤ k, denotes the minimal power of α
of the terms in the summands of sx that have the form
xi αj , where 1 ≤ i ≤ n and 0 ≤ j ≤ k − 1. The case
ri = k denotes that any term that has the form xi αj
does not occur. Again, we regard terms with larger i to
be included in the sum first in the case of repeated values
of components of x ⊗ c. Consider the sum
sy = P
n k−1
X
X
yi α j .
Case 2: rn = 0. In this case, xn ≤ Mx since xn is
included in sx . By definition of α we have xh αk−1 < Mx
and so 0 ≤ rh ≤ k − 1. Furthermore, we can prove
that rh > 0 since otherwise xh is in the summands of sx
and any terms not included are of the form xi αj where
1 ≤ i < h. By the definition of h, we have xi αj = P yi αj
for 1 ≤ i < h. So the sum of the components of x ⊗ c
not included in sx is equal to the sum of the components
of P y ⊗ c not included in sy . This fact, together with
the assumption that sx = xy will lead to a contradiction
that
1 = E1 (x ⊗ c) = P E1 (y ⊗ c) = P.
Now, if P > xh /yh , then
i=1 j=ri
It is obvious that sy ≥ P El (y ⊗ c) by definition. On the
other hand, we can rearrange the summands of sx and
sy ,respectively, as
sx =
k−1
X
j=1
α
j
n
X
xi and sy =
i=tj
k−1
X
j
α P
j=1
n
X
yi ,
i=tj
where 1 ≤ tj ≤ n + 1. Since
n
X
xi = Etj (x) ≥ P Etj (y) = P
i=tj
n
X
yi
i=tj
by the definition of P , we have sx ≥ sy . Now, if sx > sy ,
then sx > P El (y ⊗ c). In the case of sx = sy , let my
denote the minimum of the components of P y ⊗ c not
included in sy and My denote the maximum included in
sy . If we can prove my < My , then by swapping my
and My (that is, including my to and excluding My from
sy ), we can show sx > P El (y ⊗ c) again and that will
complete the proof of this theorem.
In what follows, we prove that in the assumption of
sx = sy , my < My holds. Suppose on the contrary my ≥
My ; then, sy = P El (y ⊗ c). It is then not difficult to
show mx ≥ my and Mx ≤ My , where mx and Mx are
defined analogously to my and My , since mx < my leads
to
El−1 (x ⊗ c) = sx + mx < sy + my = P El−1 (y ⊗ c)
and Mx > My leads to
El+1 (x ⊗ c) = sx − Mx < sy − My = P El+1 (y ⊗ c),
which contradict the well-known fact that P (x ⊗ c →
y ⊗ c) ≥ P (x → y).
Now, we show that a contradiction will arise by considering the following two cases.
Case 1: rn > 0. Since 1 < l < nk, we have rn < k.
Then
My ≥ Mx ≥ xn αrn > P yn αrn −1 ≥ my
since P yn /xn < α. Thus My > my , which is a contradiction.
my ≤ mx ≤ xh αrh −1 < P yh αrh ≤ My
from our assumption that α > xh /(P yh ), and if P <
xh /yh , then
My ≥ Mx ≥ xh αrh > P yh αrh −1 ≥ my
from our assumption that α > P yh /xh . Again, my ≥ My
is contradicted. That completes our proof.
Recall that in Example 1, when x = (0.6, 0.2, 0.2) and
y = (0.5, 0.3, 0.2), there exists no two-dimensional useful
catalyst for the probabilistic transformation from x to y.
We show here how to construct a higher-dimensional one
by the above theorem. It is easy to check that
P (x → y) = 0.8 <
x1
0.6
=
,
0.5
y1
so h = 1 and we need only take a real α such that
P y3 /x3 = 0.8 < α < 1 and α > P y1 /x1 = 2/3, that is,
0.8 < α < 1. In order not to make k too large, we should
take α as small as possible. For example, α = 0.801.
Then, from the constraint x3 > x1 αk−1 in the theorem,
we have k ≥ 6. Thus the state
c = (1, α, . . . , α5 )
can increase the maximal transformation probability of
x into y.
The above theorem gives us a sufficient and necessary condition under which the transformation x into y
has a catalyst which can increase the maximal probability transformation. Furthermore, the proof process constructs a real catalyst vector. What we should like to
point out here is, however, that the catalyst presented
in the proof is not very economical in the sense that it
is usually not the minimally dimensional one among all
states which can serve as a useful catalyst. How to find
a most economical one remains for further study.
Acknowledgement: This work was supported by National Foundation of Natural Sciences of China (Grant
Nos: 60273003, 60321002 and 60305005) and Key grant
Project of Chinese Ministry of Education.
5
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